Believe it or not, it’s possible to derail a train by going too slow—more about that later.

Too fast on a curve

In 1947, a Pennsylvania Railroad passenger train with 2 steam locomotives and 14 cars left Pittsburgh at 1:05 a.m. bound for New York City. The train had just descended a steep 1.73% grade when it over-turned on a sharp 8.5-degree curve (675-foot [205-m] radius) at 3:20 a.m. The speed limit downhill was 35 mph (56 km/h) and 30 mph (48 km/h) on the curve. Instructions required the train crew to test their brakes 2 miles (3.2 km) before the curve.

The 2 locomotives plunged down a 92-foot (28-m) embankment with 5 cars attached. Ten of the 14 cars derailed. Twenty-four people were killed. The investigators concluded that excess speed caused the train to overturn on the curve. The overturning speed was calculated to be 65 mph (105 km/h). Elsewhere in the news on the same day as the accident, the Pennsylvania Railroad, the largest railroad in America, reported operating losses for the year 1946—their rst ever. Speeding trains overturning on a curve also occurred in California in 1956 (killing 30) and in Virginia in 1978 (killing 6).

Inertial loading

Everyone knows, or thinks they know, what centrifugal force is. It’s the phenomenon that ings passengers against the car door on a curve, the force that keeps the water in the bucket when swung fast enough overhead, and the force that derails trains on a curve. But centrifugal force can be a source of much confusion because it’s not a force in the traditional sense. Centrifugal force is an inertial effect that occurs when a body in motion changes direction, as in each of the examples above.

Per Isaac Newton, a body in motion tends to stay in motion. If somehow we could eliminate gravity and air resistance, a ball thrown straight up would continue straight up forever. It takes additional force to change the straight-line motion of the ball and to move a train around a curve.

Inertia, the property of matter that resists changes in motion, is most easily explained by accelerating in an elevator. If a 100-lb (0.44-kN) person is standing on a scale in an elevator accelerating up, the scale reads something higher than 100 lbs. If the elevator is accelerating down, the scale reads something less than 100 lbs. If the elevator is accelerating up at 16 ft/sec2, or one-half the normal acceleration of gravity, the scale will read 150 lbs (0.66 kN). The extra 50 lbs (0.22 kN) is from the person’s body resisting acceleration.

When a body accelerates, or changes velocity, that acceleration is accompanied by a force. According to Newton’s Second Law, f m × a. The body’s inertia (m × a) is not a force even though it acts on the scale like a force. The additional 50-lb reading on the scale is the 100-lb person’s resistance to accelerating up 16 ft/sec2 (4.9 m/s2)—the person’s inertia.

Inertia always acts in the opposite direction of the acceleration. In the case of the elevator, the person is accelerating up and the inertial response is acting down and is being recorded by the scale. A similar thing happens in circular motion. Circular motion at constant speed creates an acceleration that points toward the center of rotation.

We tend to think of acceleration as being a change in speed (see Chapter 4). Velocity is actually a vector with both direction and magnitude. (The velocity vector’s magnitude is also known as speed.) Any change in

Figure 7.1

the velocity vector, be it a change in speed or a change in direction, requires a force to create the change.

Consider a ball rolling along a straight line. One could constantly tap the ball with a stick, forcing it to move in a circular path. The tapping force, always pointing to the center, is changing the ball’s velocity vector’s direction. The ball is moving at a constant speed but changing direction; the ball is said to be accelerating toward the center of the circle.

Consider a 1-lb (0.45-kg) block rotating on the end of a 4-foot (1.2-m) string in a horizontal plane at a constant speed of 20 ft/sec (6 m/s). The direction of the velocity vector, always perpendicular to the string, is constantly changing and creating acceleration toward the center of rotation (Figure 7.1).

Acceleration for circular motion equals the velocity squared divided by the radius of the circle, or 100 ft/sec2 (30.5 m/s2). Per Newton’s Law, the string must exert a force on the block equal to m × a, or a force of 3.1 lbs (13.8 N) toward the center of rotation. (Recall that to correctly calculate f m × a the weight must be converted into a mass by dividing by the acceleration of gravity—32.2 ft/sec2 [9.8 m/s2].) The force the string exerts on the block is called the centripetal, or center-seeking, force. The block exerts an inertial load on the string, keeping it tight. The so-called centrifugal force is not a force; it’s the block’s inertial resistance to the centripetal acceleration. The 1-lb (0.45-kg) block resists the acceleration imposed by the string just as the person in the elevator

Figure 7.2.

resists the upward acceleration. The term centrifugal force is incorrect. We will use the term centrifugal inertial loading. But, of course, the so-called centrifugal force feels like a force when holding on to the string attached to the rotating block.

A locomotive moving around a curve is similar to a rotating block on the end of a string. Both experience acceleration toward the center of rotation. The inertial loading keeps the string tight and creates a lateral force on the locomotive at the wheels. Lateral forces between the wheels and the rail must react against the centrifugal inertial loading to keep the train on the tracks.

If the centrifugal inertial loading is excessive, the locomotive begins to tip. The ange of the wheel catches on the rail and the locomotive starts to rotate, as shown in Figure 7.2. In fact, that’s why the anges are on the inside of the wheels. If the anges were on the outside, the slightest bit of wheel lift would slide the locomotive off the tracks.

In the 1947 Pennsylvania Railroad overturning accident, the locomotive weighed 320,000 lbs (145,150 kg). The centrifugal inertial loading of the locomotive moving at 88 ft/sec (60 mph [97km/h]) on a curve with a radius of 675 feet (206 m) is:

Horrible number photo

The centrifugal inertial loading is trying to tip the locomotive clock-wise about the pivot point (the bottom of the right wheel). This rotation is resisted by the weight of the locomotive (also acting through its center of gravity), which tries to rotate the locomotive counterclockwise.

The locomotive’s weight and inertial load both exert a torque. A torque is a twisting force applied to the end of a lever arm that tries to tighten a nut. A 10-lb (44.5-N) force on the end of a 9-inch (23-cm)-long wrench exerts a torque of 10 × 9 = 90 inch lbs of torque (10 Nm).

The Pennsylvania locomotive had a center of gravity 80 inches (2 m) above the rail. The centrifugal inertial loading tries to rotate the locomotive with a clockwise torque equal to 114,000 lbs × 80 inches—more than 9 million inch lbs of torque (6.3 × 106 Nm).

The lever arm for the locomotive’s weight is halfway between the rails, or 28 inches (0.7 m). The torque from the locomotive’s weight that tries to resist the overturning torque from the centrifugal inertial loading equals 320,000 lbs × 28 inches—almost 9 million inch lbs of torque.

The torque trying to overturn the locomotive is slightly larger than the torque from the locomotive’s weight resisting the overturning torque. The locomotive is just starting to overturn at 60 mph (97 km/h).

Superelevation

The outside rail on a curve is usually higher than the inside rail. The elevation of the outside rail relative to the inside rail is called superelevation.

A raised outside rail rotates a locomotive counterclockwise and helps ght off the clockwise rotation from the centrifugal inertial loading, at least a little bit. In fact, if the car is made top heavy and the right wheel is lifted enough (even at zero mph), eventually the car tips over counter-clockwise. The car tips over at zero mph when the weight load points outside the inner rail, as shown in Figure 7.3.

Figure 7.3

In 1947, the investigators concluded that the locomotive would over-turn on the curve (with outer rail raised or superelevated 3.5 inches [8.9 cm]) at 65 mph (105 km/h).

Amtrak’s 150-mph (241-km/h) Acela creates its own bank angle by tilting up to 4.2 degrees. If the Acela is operating on a curve whose out-side rail is raised 2 inches (5 cm), the Acela can speed as if it is on a curve that is raised an additional 7 inches (17.8 cm) higher—for a total super-elevation of 9 inches (22.9 cm).

Tilting trains are far more complicated and not the rst choice of rail-road companies. It is easier to operate on redesigned curves with larger radiuses. Of course, curves with larger radiuses take up more real estate—difcult to do in older, built-up neighborhoods.

Too fast on a turnout

Far more common is moving too fast on a turnout. On a turn-out the track crosses over with sharp turns to merge onto a parallel track. The engineer must slow the train for the turnout or risk overturning. Just such an accident happened in 1951 in New Jersey, killing 84.

Construction of the New Jersey Turnpike required relocating the train tracks 60 feet (18 m) north for a few months. The temporary track was about 2,800 feet (853 m) long and contained a 57-foot (17.4-m) temporary wooden trestle anchored on both ends by massive concrete abutments. The trestle was also part of the turnout, a 121-foot (36.9-m)-long curve with a radius of about 1,100 feet (335 m).

The speed limit on the main track was 65 mph (105 km/h). The temporary track went into operation for the rst time at one o’clock p.m. on the day of the accident, February 6, 1951. The speed limit on the turnouts and temporary track was 25 mph (40 km/h).

The rush hour train with 11 cars was particularly crowded with about 1,000 passengers, many standing. The locomotive and rst seven cars derailed. The third and fourth cars were the most damaged. Those two cars struck the concrete abutment (knocking off a big hunk) and fell down the 25-foot (7.6-m) embankment. The third car crashed onto its side, its center sill broke, and the roof and both sides were badly damaged. The right side of the fourth car was torn open its entire length. The investigators concluded that the locomotive’s speed exceeded the calculated 76 mph (122 km/h) overturning speed.

Too fast on a curve is by no means an obsolete problem. In a nearly identical accident in Chicago on September 17, 2005, a commuter train went off the tracks at a turnout, killing two. The engineer missed the signal to slow from 70 to 10 mph (113 to 16 km/h).

Too fast on a curve should be prevented in the future by Positive Train Control (see Chapter 6).

Derailing on curves

Before reaching the overturning speed, a slow, heavy freight train is far more likely to derail on a curve by rail rollover, wide gage, or wheel climb (Figure 7.4).

The tracks are constantly moving around (and constantly being readjusted) because of settlement and train forces (Figure 7.5). The track spikes do not prevent the rails from overturning but do keep them from spreading. Rail overturning is thwarted by the downward wheel forces. If the wooden cross ties are rotted, inertial loading on curves may widen the rails.

Standards are established for maximum distance between rails (gage),

Figure 7.4 and 7.5

maximum dips in each rail (prole), and maximum deviation from straightness (alignment). Higher classes of track require tighter requirements to operate safely at higher speeds. For example, freight trains are limited to 40 mph (64 km/h) on Class 3 track and 60 mph (97 km/h) on Class 4 track. (The track classes are reviewed in Chapter 11.)Although track geometry today is measured automatically with high-speed cars using laser sensors, the standards are based on low-tech methods of measuring the deviation from a 62-foot (18.8-m) string pulled tight. Every 62 feet (18.8 m) of Class 3 track can deviate up to 1.5 inches (3.8 cm) from straight and dip up to 2.25 inches (5.7 cm). The Acela operates at 150 mph (241 km/h) on Class 8 track. Every 31 feet (9.4 m) of Class 8 track can deviate up to 0.5 inch (1.27 cm) from straight and dip up to 1 inch (2.54 cm).

Class 8 track geometry is checked every 30 days. In fact, when Amtrak was preparing to operate Acela at 150 mph, Amtrak’s chief engineer of maintenance, the director of track geometry, and many others rode the geometry car every two weeks for months. They considered it a bonding experience.

The operators will also report any rough or shifted track as it occurs. For all trains operating above 125 mph (201 km/h), at least one train per day has sensors to measure, quantify, and record the location of any rough track.

Concrete, instead of wood, is used for ties on Class 8 track. The concrete is less susceptible to shifting and water damage. At least once annually Class 8 track gage stability is checked with a special car that loads the rail sideways with a force of 10,000 lbs (44.5 kN). Class 8 track is also inspected twice a year with ultrasonic sensors for internal fatigue cracks.

L/v ratios

The tendency to derail is often described by the L/V ratio, where L is the lateral force and V is the vertical force at the wheel-track interface, as shown in Figure 7.6. The higher the L/V ratio, the more likely the car is to derail.

There are rough guidelines for L/V limits. Wheel climb may occur if:

Figure 7.6

L/V greater than 1 for new freight cars with new wheels on new, straight track

L/V greater than 0.82 may be unstable on curves

L/V greater than 0.75 can be unstable for worn wheels and worn rail

L/V greater than 0.68 may overturn a poorly constrained rail

Rails spaced too close together can also encourage wheel climb.

The stated L/V ratios are merely rules of thumb, not rigid predictors. There are many other factors that interact, such as the condition of the trucks, rails, and wheels and whether or not the car body is bouncing on its suspension.

The L/V ratio can also vary greatly as the wheels and rail wear and as the contact location changes. A worn rail on the outside of a curve is

Figure 7.7

shown in Figure 7.7. Another wear pattern is shown in Figure 6.1 in Chapter 6.Acela Class 8 track must be checked annually with an instrumented car that measures the L/V ratios.3 Special load sensors are on the truck frame and on the oor of the car. If the L/V ratio exceeds 0.6, the speed must be reduced until repairs are made.

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